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lissajous

So what is a “Lissajous”? it is actually short for Lissajous curves or Lissajous figures, a class of 2D (and 3D) curves describing complex harmonic functions, or more simply multi-dimensional sine curves. The following equations describe a general Lissajous curve on an x-y coordinate plane:

x = A sin(at + φ)y = B sin(bt)

Most of the time, one leaves out the A and B, which case all the curves fall on a convenient unit square.

The most commonly described Lissajous curves set the phase term φ to π/2, i.e., a standard cosine function, and have a and b at integer ratios, like 1:2, 6:5, etc. You can think of these as natural harmonics, like in musical sounds. You can see a few of the graphs below, first for a=1 and b=2:

Here are 3:2 (a:b), and 9:8, respectively:

As you can see, the higher the ratio, the more complex and dense the figure. If you add all the figures up together, you should be able to fill the entire unit square.

There are all sorts of interesting special cases. For example, if you set a and b equal, you will get a circle. If you additionally set the φ to zero, you will get a straight line. Finally, you can mess with different values of φ, like 0.3 in the first drawing below, or set a and b to non-integer values, to get all sorts of interesting variations:

It is interesting to think about these sorts of functions by relating them both visually and aurally (i.e., synthesizing the corresponding waveforms), but we will leave that as an exercise for interested readers, perhaps returning to the topic in a future article.